Hutch++: Optimal Stochastic Trace Estimation

We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \pm \epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/\epsilon)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/\epsilon^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory requires $A$ to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.

Model-specific Data Subsampling with Influence Functions

Model selection requires repeatedly evaluating models on a given dataset and measuring their relative performances. In modern applications of machine learning, the models being considered are increasingly more expensive to evaluate and the datasets of interest are increasing in size. As a result, the process of model selection is time-consuming and computationally inefficient. In this work, we develop a model-specific data subsampling strategy that improves over random sampling whenever training points have varying influence. Specifically, we leverage influence functions to guide our selection strategy, proving theoretically, and demonstrating empirically that our approach quickly selects high-quality models.

Subspace Embeddings Under Nonlinear Transformations

We consider low-distortion embeddings for subspaces under \emph{entrywise nonlinear transformations}. In particular we seek embeddings that preserve the norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where $Z$ is a $k$-dimensional subspace of $\mathbb{R}^n$ and $f(x)$ is a nonlinear activation function applied entrywise to $x$. When $f$ is the identity, and so $S$ is just a $k$-dimensional subspace, it is known that, with high probability, a random embedding into $O(k/\epsilon^2)$ dimensions preserves the norm of all $y \in S$ up to $(1\pm \epsilon)$ relative error. Such embeddings are known as \emph{subspace embeddings}, and have found widespread use in compressed sensing and approximation algorithms. We give the first low-distortion embeddings for a wide class of nonlinear functions $f$. In particular, we give additive $\epsilon$ error embeddings into $O(\frac{k\log (n/\epsilon)}{\epsilon^2})$ dimensions for a class of nonlinearities that includes the popular Sigmoid SoftPlus, and Gaussian functions. We strengthen this result to give relative error embeddings under some further restrictions, which are satisfied e.g., by the Tanh, SoftSign, Exponential Linear Unit, and many other `soft' step functions and rectifying units. Understanding embeddings for subspaces under nonlinear transformations is a key step towards extending random sketching and compressing sensing techniques for linear problems to nonlinear ones. We discuss example applications of our results to improved bounds for compressed sensing via generative neural networks.

Fourier Sparse Leverage Scores and Approximate Kernel Learning

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study $s$-sparse functions of the form $f(x) = \sum_{j=1}^s a_j e^{i \lambda_j x}$ for coefficients $a_j \in \mathbb{C}$ and frequencies $\lambda_j \in \mathbb{R}$. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the $statistical\ dimension$ of the approximated kernel matrix. It is the first "oblivious sketching method" with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.

Node Embeddings and Exact Low-Rank Representations of Complex Networks

Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such embeddings cannot capture local structure arising in complex networks. In particular, they show that any network generated from a natural low-dimensional model cannot be both sparse and have high triangle density (high clustering coefficient), two hallmark properties of many real-world networks. In this work we show that the results of Seshadhri et al. are intimately connected to the model they use rather than the low-dimensional structure of complex networks. Specifically, we prove that a minor relaxation of their model can generate sparse graphs with high triangle density. Surprisingly, we show that this same model leads to exact low-dimensional factorizations of many real-world networks. We give a simple algorithm based on logistic principal component analysis (LPCA) that succeeds in finding such exact embeddings. Finally, we perform a large number of experiments that verify the ability of very low-dimensional embeddings to capture local structure in real-world networks.

InfiniteWalk: Deep Network Embeddings as Laplacian Embeddings with a Nonlinearity

The skip-gram model for learning word embeddings (Mikolov et al. 2013) has been widely popular, and DeepWalk (Perozzi et al. 2014), among other methods, has extended the model to learning node representations from networks. Recent work of Qiu et al. (2018) provides a closed-form expression for the DeepWalk objective, obviating the need for sampling for small datasets and improving accuracy. In these methods, the "window size" T within which words or nodes are considered to co-occur is a key hyperparameter. We study the objective in the limit as T goes to infinity, which allows us to simplify the expression of Qiu et al. We prove that this limiting objective corresponds to factoring a simple transformation of the pseudoinverse of the graph Laplacian, linking DeepWalk to extensive prior work in spectral graph embeddings. Further, we show that by a applying a simple nonlinear entrywise transformation to this pseudoinverse, we recover a good approximation of the finite-T objective and embeddings that are competitive with those from DeepWalk and other skip-gram methods in multi-label classification. Surprisingly, we find that even simple binary thresholding of the Laplacian pseudoinverse is often competitive, suggesting that the core advancement of recent methods is a nonlinearity on top of the classical spectral embedding approach.

Efficient Intervention Design for Causal Discovery with Latents

We consider recovering a causal graph in presence of latent variables, where we seek to minimize the cost of interventions used in the recovery process. We consider two intervention cost models: (1) a linear cost model where the cost of an intervention on a subset of variables has a linear form, and (2) an identity cost model where the cost of an intervention is the same, regardless of what variables it is on, i.e., the goal is just to minimize the number of interventions. Under the linear cost model, we give an algorithm to identify the ancestral relations of the underlying causal graph, achieving within a $2$-factor of the optimal intervention cost. This approximation factor can be improved to $1+\epsilon$ for any $\epsilon > 0$ under some mild restrictions. Under the identity cost model, we bound the number of interventions needed to recover the entire causal graph, including the latent variables, using a parameterization of the causal graph through a special type of colliders. In particular, we introduce the notion of $p$-colliders, that are colliders between pair of nodes arising from a specific type of conditioning in the causal graph, and provide an upper bound on the number of interventions as a function of the maximum number of $p$-colliders between any two nodes in the causal graph.

Projection-Cost-Preserving Sketches: Proof Strategies and Constructions

In this note we illustrate how common matrix approximation methods, such as random projection and random sampling, yield projection-cost-preserving sketches, as introduced in [FSS13, CEM+15]. A projection-cost-preserving sketch is a matrix approximation which, for a given parameter $k$, approximately preserves the distance of the target matrix to all $k$-dimensional subspaces. Such sketches have applications to scalable algorithms for linear algebra, data science, and machine learning. Our goal is to simplify the presentation of proof techniques introduced in [CEM+15] and [CMM17] so that they can serve as a guide for future work. We also refer the reader to [CYD19], which gives a similar simplified exposition of the proof covered in Section 2.

Importance Sampling via Local Sensitivity

Given a loss function $F:\mathcal{X} \rightarrow \mathbb{R}^+$ that can be written as the sum of losses over a large set of inputs $a_1,\ldots, a_n$, it is often desirable to approximate $F$ by subsampling the input points. Strong theoretical guarantees require taking into account the importance of each point, measured by how much its individual loss contributes to $F(x)$. Maximizing this importance over all $x \in \mathcal{X}$ yields the \emph{sensitivity score} of $a_i$. Sampling with probabilities proportional to these scores gives strong provable guarantees, allowing one to approximately minimize of $F$ using just the subsampled points. Unfortunately, sensitivity sampling is difficult to apply since 1) it is unclear how to efficiently compute the sensitivity scores and 2) the sample size required is often too large to be useful. We propose overcoming both obstacles by introducing the \emph{local sensitivity}, which measures data point importance in a ball around some center $x_0$. We show that the local sensitivity can be efficiently estimated using the \emph{leverage scores} of a quadratic approximation to $F$, and that the sample size required to approximate $F$ around $x_0$ can be bounded. We propose employing local sensitivity sampling in an iterative optimization method and illustrate its usefulness by analyzing its convergence when $F$ is smooth and convex.